Statistica Sinica 20 (2010), 235-238

NECESSARY AND SUFFICIENT CONDITIONS FOR

CONSISTENCY OF A METHOD FOR

SMOOTHED FUNCTIONAL INVERSE REGRESSION

R. D. Cook, L. Forzani and A. F. Yao

University of Minnesota, Instituto de Matemática Aplicada del Litoral

and Université de la Méditerranée

Abstract: Ferré and Yao (2005, 2007) proposed a method to estimate the Effective Dimension Reduction space in functional sliced inverse regression. Their approach did not require the inversion of the variance-covariance operator of the explanatory variables, and it allowed them to get $\sqrt{n}$ consistent estimators in the functional case. In those papers there is a mistake. In this note we show that, in general, the approach does not give an estimator of the SIR subspace. We also give necessary and sufficient conditions for this to be true.

Key words and phrases: Dimension reduction, functional data analysis, inverse regression.